## The number of $$k$$-undivided families of subsets of an $$n$$-element set ($$k$$-undivided Boolean functions). I: The case when $$n$$ is even and $$k=2$$.(Russian)Zbl 1032.05006

Summary: Let $$S$$ be a finite set that consists of $$n$$ different elements and $$k\geq 2$$ be a natural number. A family $${\mathcal F}$$ of subsets $$S_1,\dots,S_r$$, $$r\geq k$$, of the set $$S$$ is called $$k$$-undivided if the intersection of any $$k$$ sets of $${\mathcal F}$$ is non-empty. Such families are equivalent to $$k$$-undivided Boolean functions of $$n$$ variables, i.e. to functions $$f(x_1,\dots,x_n)$$ such that any $$k$$ vectors with $$f(x_1,\dots,x_n)=1$$ have at least one component equal to 1. In the paper, an asymptotic for the size of a special subset of $$2$$-undivided Boolean functions of $$n$$ variables, where $$n$$ is even, is given. A proof of the fact that this asymptotic coincides with an asymptotic for the number of all $$2$$-undivided Boolean functions of $$n$$ variables will be presented in the next paper.

### MSC:

 05A16 Asymptotic enumeration 03E05 Other combinatorial set theory 06E30 Boolean functions

### Keywords:

two-valued function; asymptotic expression; Post class